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3.1 Geometry

General relativity in its canonical formulation [6] describes the geometry of space-time in terms of fields on spatial slices. Geometry on such a spatial slice S is encoded in the spatial metric qab, which presents the configuration variables. Canonical momenta are given in terms of extrinsic curvature Kab which is the derivative of the spatial metric under changing the spatial slice. Those fields are not arbitrary since they are obtained from a solution of Einstein’s equations by choosing a time coordinate defining the spatial slices, and space-time geometry is generally covariant. In the canonical formalism this is expressed by the presence of constraints on the fields, the diffeomorphism constraint and the Hamiltonian constraint. The diffeomorphism constraint generates deformations of a spatial slice or coordinate changes, and when it is satisfied spatial geometry does not depend on which coordinates we choose on space. General covariance of space-time geometry also for the time coordinate is then completed by imposing the Hamiltonian constraint. This constraint, furthermore, is important for the dynamics of the theory: Since there is no absolute time, there is no Hamiltonian generating evolution, but only the Hamiltonian constraint. When it is satisfied, it encodes correlations between the physical fields of gravity and matter such that evolution in this framework is relational. The reproduction of a space-time metric in a coordinate dependent way then requires to choose a gauge and to compute the transformation in gauge parameters (including the coordinates) generated by the constraints.

It is often useful to describe spatial geometry not by the spatial metric but by a triad a ei which defines three vector fields which are orthogonal to each other and normalized in each point. This yields all information about spatial geometry, and indeed the inverse metric is obtained from the triad by qab = eaeb i i where we sum over the index i counting the triad vector fields. There are differences, however, between metric and triad formulations. First, the set of triad vectors can be rotated without changing the metric, which implies an additional gauge freedom with group SO(3) acting on the index i. Invariance of the theory under those rotations is then guaranteed by a Gauss constraint in addition to the diffeomorphism and Hamiltonian constraints.

The second difference will turn out to be more important later on: We can not only rotate the triad vectors but also reflect them, i.e., change the orientation of the triad given by a sgndet ei. This does not change the metric either, and so could be included in the gauge group as O(3). However, reflections are not connected to the unit element of O(3) and thus are not generated by a constraint. It then has to be seen whether or not the theory allows to impose invariance under reflections, i.e., if its solutions are reflection symmetric. This is not usually an issue in the classical theory since positive and negative orientations on the space of triads are separated by degenerate configurations where the determinant of the metric vanishes. Points on the boundary are usually singularities where the classical evolution breaks down such that we will never connect between both sides. However, since there are expectations that quantum gravity may resolve classical singularities, which indeed are confirmed in loop quantum cosmology, we will have to keep this issue in mind and not restrict to only one orientation from the outset.


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